In the field of evolutionary algorithms, basic principles of natural evolution are used for generating or optimizing technical structures. Basic operations are mutation and recombination as a method for modifying structures or parameters. To eliminate unfavorable modifications and to proceed with modifications which increase the overall quality of the system, a selection operation is used. Principles of the evolution strategy can be found for example in Rechenberg, Ingo (1994) “Evolutionsstrategie”, Friedrich Frommann Holzboog Verlag, which is incorporated by reference herein in its entirety.
The application of EAs in the optimization of designs is well known, see for example the book “Evolutionary Algorithms in Engineering Applications” by D. Dasgupta and Z. Michalewicz, Springer Verlag, 1997, which is incorporated by reference herein in its entirety.
As evolutionary algorithms are more and more successfully used as optimization tools for large-scale “real-world” problems, the influence of noise on the performance and the convergence properties of evolutionary algorithms have come into focus.
Quality evaluations in optimization processes are frequently noisy due to design uncertainties concerning production tolerances or actuator imprecision acting directly on the design variables x. That is, the performance ƒ of a design becomes a stochastic quantity {tilde over (ƒ)} via internal design perturbations{tilde over (ƒ)}(x)=ƒ(x+δ), δ-random vector,where the random vector δ obeys a certain unknown distribution (often modeled as a Gaussian distribution) andE[δ]=0.
This means, given a design x, evaluating its quality {tilde over (ƒ)}(x) necessarily yields stochastic quantity values. As a result, an optimization algorithm applied to {tilde over (ƒ)}(x) must deal with these uncertainties and it must use this information to calculate a robust optimum based on an appropriate robustness measure.
Probably the most widely used measure is the expected value of {tilde over (ƒ)}(x), that is
      E    ⁡          [                        f          ~                ⁢                  |                ⁢        x            ]        .Assuming a continuous design space, the expected value robustness measure is given by the integral
      E    ⁡          [                        f          ~                ⁢                  |                ⁢        x            ]        =            ∫              ℜ        N              ⁢                  f        ⁡                  (                      x            +            δ                    )                    ⁢              p        ⁡                  (          δ          )                    ⁢                          ⁢                        ⅆ          N                ⁢        δ            and the optimal design x is formally obtained by
      x    ^    =      arg    ⁢                  ⁢          opt      x        ⁢                  ∫                  ℜ          N                    ⁢                        f          ⁢                      (                          x              +              δ                        )                          ⁢                  p          ⁡                      (            δ            )                          ⁢                                  ⁢                                            ⅆ              N                        ⁢            δ                    .                    
If one were able to calculate
  E  ⁡      [                  f        ~            ⁢              |            ⁢      x        ]  analytically, the resulting optimization problem would be an ordinary one, and standard (numerical) optimization techniques could be applied. However, real-world applications will usually not allow for an analytical treatment, therefore one has to rely on numerical estimates of
  E  ⁡      [                  f        ~            ⁢              |            ⁢      x        ]  using Monte-Carlo simulations. Alternatively one can use direct search strategies capable of dealing with the noisy information directly.
The latter is the domain of evolutionary algorithms (EAs). In particular, evolutionary algorithms have been shown to cope with such stochastic variations better than other optimization algorithms, see e.g. “On the robustness of population-based versus point-based optimization in the presence of noise” by V. Nissen and J. Propach, IEEE Transactions on Evolutionary Computation 2(3):107-119, 1998, which is incorporated by reference herein in its entirety.
A conventional technique to find approximate solutions to the above equation using EAs is to use the design uncertainties δ explicitly. That is, given an individual design x, the perturbation δ is explicitly added to the design x. While the EA works on the evolution of x, the goal function in the black-box is evaluated with respect to {tilde over (x)}:=x+δ. Since in center of mass evolution strategies an individual offspring design is the result of a mutation z applied to a parental individual and the parental centroid x), respectively, the actually design tested is{tilde over (x)}:=x+z+δ. 
Taking now another perspective, one might interpret z+δ as a mutation in its own right. This raises the question whether it is really necessary to artificially add the perturbation in a black-box to the design x. As an alternative one might simply use a mutation {tilde over (z)}=z+δ with a larger mutation strength instead of z. In other words, the mutations themselves may serve as robustness tester.
However, even though evolutionary algorithms/strategies are regarded as well suited for noisy optimization, its application to robust optimization bears some subtleties/problems: due to selection, the robustness of a design x is not tested with respect to samples of the density function p(δ). Selection prefers those designs which are by chance well adopted to the individual realizations of the perturbation δ.
For example, when considering actuator noise of standard deviation ε on a sphere model ∥x∥2 (to be minimized), the actually measured standard deviation Di of a specific component i of the parent population will usually be smaller, i.e. Di<σ. This is so, because selection singles out all those x+δ states with large length ∥x+δ∥. That is, shorter δ vectors are preferred resulting in a smaller measured standard deviation. Therefore, an evolutionary algorithm for robust optimization must take into account this effect and take appropriate counter measures.
Furthermore, it is well known that noise deteriorates the performance of the evolutionary algorithms. If the function to be optimized is noisy at its global or local optimizer, the evolutionary algorithm cannot reach the optimizer in expectation. That is, the parental individuals are located in the long run (steady state behavior) in a certain (expected) distance to the optimizer, both in the object parameter space and usually also in the quality/fitness space.
What is needed is an improved system and method (1) for evaluating the robustness of an Evolutionary Algorithm; (2) where the observed parental variance is controlled such that robustness (with regard to noise etc.) is tested correctly; and/or (3) for optimization that is driven by the trade-off between reducing the residual distance (induced by the noise) to the optimizer state and reducing the number of required fitness evaluations. In other words, such a method for optimization can reduce the residual distance (induced by the noise) to the optimizer state while at the same time minimizing the required additional fitness evaluation effort.